Dwight argued they worshipped different gods.
Should it be said in opposition to the observations, which I have made concerning the intelligibleness of the Scriptures, that my antagonists will grant, that the Scriptures are thus plain, in points of essential importance to our duty and salvation ; but need not be supposed to be so in mere speculative opinions: I answer, that no doctrine is of more importance, whether speculative or practical, than that, which teaches the character of Christ; except that, which teaches the existence and perfections of God. If Christ be a creature; all the worship, and all other regard, rendered to him as the Creator, is unquestionably mere Idolatry: the sin, which of all sins is the most strongly threatened, and reproved, in the Scriptures. If Christ is God; then a denial that he is God, is all that is meant by impiety. It is a denial of his primary and essential Character; of the Attributes, which in this character belong to him; of the Relations, which he sustains to the Universe, and will for ever sustain; of the actions, which he has performed, and will perform throughout eternity; and of the essential glory, which he had with the Father before ever the world was. Man is a being, made up of an animal body and a rational mind. Should I deny, that a particular person possessed a rational mind; would it not be justly said, that I denied him to be a man, and refused to acknowledge his primary and most essential character? If Christ is God-man; and I deny him to be God ; do I not, at least as entirely, deny his primary and most essential character? In other words, do I not plainly deny the Lord that bought me? It is evidently impossible for him, who makes this denial, to render to Christ those regards ; that confidence, love, reverence, and obedience; which a man, who believed Christ to be God, would feel himself indispensably bound to render. Indeed, were it possible, he would necessarily, and in the very act of rendering them, condemn himself as guilty of Idolatry. On the other hand, he, who believes Christ to be God, cannot refuse to render them, without condemning himself as guilty, and without being actually guilty, of the plainest and grossest impiety ; because he withholds from the true God, the homage and obedience, due to his character. The Unitarians censure the system of the Trinitarians as being idolatrous, and them as being Idolaters. If the Unitarian scheme is true, the censure is just. We, on the other hand, and with equal justice, if our scheme is true, declare them to be guilty of direct and gross impiety; because they worship not the Father, the Son, and the Holy Ghost; the Jehovah of the Scriptures; the Jehovah Aleim, who is one Jehovah ; but another and very different God.
The admission of the Deity of Christ, therefore, if he really be God, is a fundamental doctrine of Christianity; mistakes about which are altogether dangerous and dreadful....
So this is how, according to Timothy Dwight's logic, orthodox Trinitarian Christians OUGHT to view the God of the American Founding, the God of the Declaration of Independence. The God of the American Founding is not necessarily or identifiably Trinitarian. The biblical God is necessarily and identifiably Trinitarian. Therefore, the God of the American Founding is not necessarily or identifiably the God of the Bible.
Unitarians, though perhaps not a majority of the FFs, did play disproportionate roles during the American Founding. Jefferson, J. Adams and Franklin wrote the Declaration, for instance. Even though they did not (and could not) publicly declare the God of the American Founding as non-Triune, the lack of identifiably Trinitarian attributes of the God of America's Founding civil religion (for which I DO see the unitarians as responsible) arguably makes that God implicitly unitarian (and hence, according to Trinitarians, not the God of the Bible).